Advertisements
Advertisements
Question
Differentiate \[\tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Solution
\[\text{Let, y } = \tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}\]
\[\text{ Put x } = a \sin\theta\]
\[ y = \tan^{- 1} \left\{ \frac{a \sin\theta}{\sqrt{a^2 - a^2 \sin^2 \theta}} \right\}\]
\[ y = \tan^{- 1} \left( \frac{a \sin\theta}{\sqrt{a^2 \left( 1 - \sin^2 \theta \right)}} \right) \]
\[ y = \tan^{- 1} \left\{ \frac{a \sin\theta}{a \cos\theta} \right\} \]
\[ y = \tan^{- 1} \left( \tan\theta \right) ...........\left( 1 \right)\]
\[Here, - a < x < a\]
\[ \Rightarrow - 1 < \frac{x}{a} < 1\]
\[ \Rightarrow \sin\left( - \frac{\pi}{2} \right) < \sin\theta < \sin\left( \frac{\pi}{2} \right) \left( \because x = a \sin\theta \right)\]
\[ \Rightarrow - \frac{\pi}{2} < \theta < \frac{\pi}{2}\]
\[\text{ So, from equation } \left( 1 \right), \]
\[ y = \theta \left[ \text{ Since}, \tan^{- 1} \left( \tan\theta \right) = \theta, \text{ if } \theta \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right] \right]\]
\[ y = \sin^{- 1} \left( \frac{x}{a} \right) \left[ \text{ Since, x } = a \sin\theta \right]\]
\[\text{ Differentiating it with respect to x }, \]
\[\text{ Using chain rule }, \]
\[\frac{d y}{d x} = \frac{1}{\sqrt{1 - \left( \frac{x}{a} \right)^2}}\frac{d}{dx}\left( \frac{x}{a} \right)\]
\[\frac{d y}{d x} = \frac{a}{\sqrt{a^2 - x^2}} \times \left( \frac{1}{a} \right)\]
\[\frac{d y}{d x} = \frac{1}{\sqrt{a^2 - x^2}}\]
APPEARS IN
RELATED QUESTIONS
Differentiate sin (3x + 5) ?
Differentiate \[3^{e^x}\] ?
Differentiate \[3^{x^2 + 2x}\] ?
Differentiate \[3^{x \log x}\] ?
If \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
If \[\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}\] ?
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[\left( \log x \right)^{ \log x }\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
If \[xy \log \left( x + y \right) = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
If \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?
If \[y = \sqrt{\cos x + \sqrt{\cos x + \sqrt{\cos x + . . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sin x}{1 - 2 y}\] ?
Find \[\frac{dy}{dx}\] when \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\] ?
If f (x) = loge (loge x), then write the value of `f' (e)` ?
If \[u = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ and v} = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] where \[- 1 < x < 1\], then write the value of \[\frac{du}{dv}\] ?
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
If \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
Find the minimum value of (ax + by), where xy = c2.
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
f(x) = xx has a stationary point at ______.
